If $|f(z)|\le |g(z)|$, can we say $f$ is constant?

Question

Let $f$ and $g$ be entire functions and $g(z)\neq 0$ for all $z\in \mathbb{C}$. If $|f(z)|\le |g(z)|$, can we say $f$ is constant?

Liouville theorem says that an entire bounded function is constant, but g is not given bounded here. So I think it should be false. What else can we say about $f$?

What you have is that $f/g$ is bounded. Thus $f/g$ is constant. That is, $g=cf$ for some real constant $c$ with $|c|\ge 1.$ — mfl, Dec 20 '16 at 19:20
In any case, you have $\phi = {f \over g}$ is entire and $|\phi(z)| \le 1$ for all $z$, hence $\phi$ is constant. Bingo. — copper.hat, Dec 20 '16 at 19:20
You are right in supposing $f$ need not be constant: Let $g$ be any unbounded entire function and $f(z) = g(z)/2$. Then $f(z)$ is entire but non-constant. — Mark Fischler, Dec 20 '16 at 19:21

score 6 · Answer 1 · answered Dec 20 '16 at 19:20

6

$\frac{f(z)}{g(z)}$ is entire and bounded => $\frac{f(z)}{g(z)}$ is constant => f(z) = c*g(z)

answered Dec 20 '16 at 19:20

kotomord

1,814

score 3 · Answer 2 · answered Dec 21 '16 at 01:52

3

No, we cannot conclude that $f$ is constant. Here's a counter-example:

If $f(z) = e^z$ and $g(z) = 2f(z) = 2e^z$. Clearly $f$ and $g$ are both entire and non-constant. We also see that $|f(z)| \leq |g(z)|$ for all $z$ and $g(z) \neq 0$.

answered Dec 21 '16 at 01:52

ChocolateAndCheese

1,542

score 2 · Answer 3 · edited Apr 13 '17 at 12:21

2

If this holds then $f = e^{\phi(z)}$ for some entire $\phi$.

Using Weierstrass Factorisation Theorem it can be shown that $g = e^{\psi(z)}$ for some entire $\psi$. (See this)

As copper.hat pointed out, $f/g$ must be constant, so $f$ must also be in the form $e^{\phi(z)}$.

edited Apr 13 '17 at 12:21

Community

1

answered Dec 20 '16 at 19:21

Henricus V.

18,694

That's a big weapon to use here. – zhw. Dec 23 '16 at 18:40

If $|f(z)|\le |g(z)|$, can we say $f$ is constant?

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